Helicity evolution at smallx: Flavor singlet and nonsinglet observables

We extend our earlier results for the quark helicity evolution at small x J. High Energy Phys. 01 (2016) 072 to derive the small-x asymptotics of the flavor singlet and flavor nonsinglet quark helicity TMDs and PDFs and of the g₁ structure function. In the flavor singlet case we rederive the evolution equations obtained in our previous paper on the subject J. High Energy Phys. 01 (2016) 072, performing additional cross-checks of our results. In the flavor nonsinglet case we construct new small-x evolution equations by employing the large-N₂ limit. All evolution equations resum double-logarithmic powers of ₒln^2 (1/x) in the polarization-dependent evolution along with the single-logarithmic powers of ₒln (1/x) in the unpolarized evolution which includes saturation effects. We solve the linearized flavor nonsinglet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin Z. Phys. C 70, 273 (1996) using the infrared evolution equations. Our numerical solution of the linearized large-N₂ evolution equations for the flavor singlet case is presented in the accompanying Letter Phys. Rev. Lett. 118, 052001 (2017) and is further discussed here.